A new class of communications equipment has developed in which a communications signal may be processed using digitally implemented circuitry. The present invention relates to the generation of a digital signal which comprises samples of the complex sinusoid: EQU w(t)=e.sup.j2.pi.f.sbsp.c.sup.t
where f.sub.c is the desired oscillator frequency.
The above described digital signal may be advantageously utilized for quadrature mixing operations performed in a zero intermediate frequency (I.F.) section of a digitally implemented receiver. According to conventional communications theory, EQU e.sup.j2.pi.f.sbsp.c.sup.t =cos 2.pi.f.sub.c t+j sin 2.pi.f.sub.c t
Therefore, the generation of cosine and sine waveforms are special cases (real and imaginary parts, respectively) of the more general complex sinusoid generation. The sampled version of e.sup.j2.pi.f.sbsp.c.sup.t is obtained by replacing the continuous time variable t by a digital discrete time variable nT, where n is a counting integer (1,2,3, . . . ) and T is the sampling period, which equals 1/fs=1/sampling rate. The discrete time signal is then equivalent to: EQU w(n)=e.sup.j2.pi.f.sbsp.c.sup.(nT)
ROM lookup methods of generating this signal follow from making the frequency variable f.sub.c, as well as the time variable (nT), discrete. If we let f.sub.c =kfs/2.sup.N (where k is an integer), then: EQU w(n)=e.sup.j2.pi.kf.sbsp.s.sup.(n/f.sbsp.s.sup.)/2.spsp.N =e.sup.j2.pi.nk/.sup.2.spsp.N
The frequency resolution obtained for the digitally implemented oscillator is equivalent to: EQU .DELTA.f=f.sub.s /2.sup.N
wherein 2.sup.N distinct frequencies can be generated.
One type of digitally implemented oscillator provides separate cosine and sine ROM tables, each with 2.sup.N words. By exploiting sine-cosine symmetries and allowing two lookups, one for cosine and one for sine, the amount of ROM may be reduced to 1/4.multidot.2.sup.N words. That is, either the cosine or sine values for only one quarter of the circle need be stored.
The above mentioned technique is referred to as a direct ROM Lookup. The amount of required ROM may be further reduced by employing a technique referred to as factored ROM lookup. This method provides that the complex phasor e.sup.j.phi. may be broken into 2 or more factors, each with its own lookup table, whose product is calculated in real time using complex digital multiplications. Therefore, the ROM requirement may be reduced at the expense of the need to perform complex digital multiplications. For example, the phasor e.sup.j.phi. can be broken into two factors, coarse and fine according to the following equation: EQU e.sup.j.phi. =e.sup.j.phi.c .multidot.e.sup.j.phi.f
If .phi. is represented in, say 16 bits, Direct ROM lookup requires at least 1/4.multidot.2.sup.16 or 16,384 words of ROM.
In Factored ROM Lookup, the 16-bit integer representing .phi. can be separated into an 8-bit coarse part (most significant bits) and an 8-bit fine part (least significant bits), which address coarse and fine ROM tables, respectively. Therefore, the amount of ROM is reduced to approximately 2.times.2.sup.8 or 512 words, at the expense of one complex digital multiply operation. Further discussion of the background of digital oscillators is set forth in an article by J. Tierney et al., entitled "A Digital Frequency Synthesizer", IEEE Trans. Audio and Electroacoustics, March 1971.
An essential characteristic of the conventionally implemented ROM lookup oscillators described above is that as the frequency resolution is increased, the ROM size requirement also increases. For the Direct method, each doubling of resolution leads to doubling of the ROM size. For the Factored method with two factors, doubling the resolution increases the ROM size by a factor of about .sqroot.2.